Keep in mind that there is in truth no central core theory of nonlinea — Partial differential equation

"One morning early in my (Hershs) years as a thesis student of Peter Lax, I entered my mentors office to find him glowing in smiles. “Louis is back!” he cried out to me. Louis, I wondered? Oh yes, Louis Nirenberg, also one of the partial differential specialists on the faculty of NYUs Courant Institute. He had been on leave in England; now he was back home! At the time. I didnt get it. Louis Nirenberg and Peter Lax were grad students together at NYU. Then they both stayed on to become famous faculty members there—Louis, a world master at elliptic partial differential equations, and Peter, a world master of hyperbolic PDEs. They hardly ever collaborated or produced joint publications. But their conversations and their intellectual and emotional interactions were a vital part of their creativity and success."

"The first systematic attack on a problem involving a partial differential equation was carried out in a sequence of 1746 papers by Jean Le Rond dAlembert (1717-1783), who sought the fundamental modes of vibration of a vibrating string."

"The development of the theory of P.D.E. is closely linked with advances in complex analysis; in fact, Riemann’s approach to the study of conformal mapping via the Dirichlet principle led to the systematic development of the theory of elliptic P.D.E. and associated variational problems. The application of these methods to the theory of several complex variables was initiated by Hodge in his theory of harmonic integrals on compact manifolds. It is this work that led H. Weyl to prove the fundamental hypoellipticity theorem, known as Weyl’s lemma, which in turn led to the development of the general theory of elliptic P.D.E."

"If the original work of Calderón and Zygmund was related to elliptic PDEs, later developments allowed applications of their theory to parabolic equations and to general hypoelliptic operators, and the more recent explosion of interest in the theory of oscillatory integrals and in problems involving curvature has much to do with hyperbolic equations."